new progress on factorized groups and subgroup permutability · new progress on factorized groups...

New progress on factorized groups andsubgroup permutability

Paz Arroyo-Jordá

Instituto Universitario de Matemática Pura y AplicadaUniversidad Politécnica de Valencia, Spain

Groups St Andrews 2013 in St AndrewsSt Andrews, 3rd-11th August 2013

in collaboration with M. Arroyo-Jordá, A. Martínez-Pastor and M.D. Pérez-Ramos

P. Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 1 / 30

Introduction General problem

Factorized groups

All groups considered will be finite

Factorized groups: A and B subgroups of a group G

G = AB

How the structure of the factors A and B affects the structure ofthe whole group G?How the structure of the group G affects the structure of A and B?


Introduction General problem

Factorized groups

Natural approach: Classes of groups

A class of groups is a collection F of groups with the property that if G ∈ Fand G ∼= H, then H ∈ F

Question

Let F be a class of groups and G = AB a factorized group:A,B ∈ F =⇒ G ∈ F?G ∈ F =⇒ A,B ∈ F?


Introduction Formations

Definitions

A formation is a class F of groups with the following properties:Every homomorphic image of an F-group is an F-group.If G/M and G/N ∈ F , then G/(M ∩ N) ∈ F

F a formation: the F-residual GF of G is the smallest normalsubgroup of G such that G/GF ∈ F

The formation F is said to be saturated if G/Φ(G) ∈ F , thenG ∈ F .


Introduction Products of supersoluble groups

Starting point

G = AB : A,B ∈ U , A,B E G 6=⇒ G ∈ U

Example

Q = 〈x , y〉 ∼= Q8, V = 〈a,b〉 ∼= C5 × C5G = [V ]Q the semidirect product of V with QG = AB with A = V 〈x〉 and B = V 〈y〉A,B ∈ U , A,B E G, G 6∈ U

G = AB : A,B ∈ U , A,B E G + additional conditions =⇒ G ∈ U

(Baer, 57) G′ ∈ N(Friesen,71) (|G : A|, |G : B|) = 1


Introduction Products of supersoluble groups

Permutability properties

If G = AB is a central product of the subgroups A and B, then:A,B ∈ U =⇒ G ∈ U

More generally, if F is any formation:A,B ∈ F =⇒ G ∈ F

(In particular, this holds when G = A× B is a direct product.)

Let G = AB a factorized group:

A,B ∈ U ( or F )+ permutability properties =⇒ G ∈ U ( or F )


Permutability properties Total permutability

Total permutability

Definition

Let G be a group and let A and B be subgroups of G. It is said that Aand B are totally permutable if every subgroup of A permutes withevery subgroup of B.

Theorem

(Asaad,Shaalan, 89) If G = AB is the product of the totally permutablesubgroups A and B, then

A,B ∈ U =⇒ G ∈ U


Permutability properties Total permutability

Total permutability and formations

(Maier,92; Carocca,96;Ballester-Bolinches, Pedraza-Aguilera, Pérez-Ramos,96-98) Let F be a formation such that U ⊆ F . Let the groupG = G1G2 · · ·Gr be a product of pairwise totally permutable subgroupsG1,G2, . . . ,Gr , r ≥ 2. Then:

Theorem

If Gi ∈ F ∀i ∈ {1, . . . , r}, then G ∈ F .

Assume in addition that F is either saturated or F ⊆ S. If G ∈ F ,then Gi ∈ F , ∀i ∈ {1, . . . , r}.

Corollary

If F is either saturated or F ⊆ S, then: GF = GF1 GF2 . . .GFr .


Permutability properties Conditional permutability

Conditional permutability

Definitions

(Qian,Zhu,98) (Guo, Shum, Skiba, 05) Let G be a group and let A and B besubgroups of G.

A and B are conditionally permutable in G (c-permutable), if ABg = BgAfor some g ∈ G.A and B are totally c-permutable if every subgroup of A is c-permutablein G with every subgroup of B.

Permutable =⇒6⇐=

Conditionally Permutable

Example

Let X and Y be two 2-Sylow subgroups of S3. Then X permutes with Y g forsome g ∈ S3, but X does not permute with Y .



Total c-permutability and supersolubility

Theorem

(Arroyo-Jordá, AJ, Martínez-Pastor,Pérez-Ramos,10) Let G = AB be theproduct of the totally c-permutable subgroups A and B. Then:

GU = AUBU

In particular, A,B ∈ U ⇐⇒ G ∈ U

Corollary

(AJ, AJ, MP, PR, 10) Let G = AB be the product of the totallyc-permutable subgroups A and B and let p be a prime. If A,B arep-supersoluble, then G is p-supersoluble.



Total c-permutability and saturated formations

Question

Are saturated formations F (of soluble groups) containing U closed undertaking products of totally c-permutable subgroups?

Example

Take G = S4 = AB, A = A4 and B ∼= C2 generated by a transposition. Then Aand B are totally c-permutable in G.Let F = N 2, the saturated formation of metanilpotent groups. NoticeU ⊆ N 2 ⊆ S. Then:

A,B ∈ F but G 6∈ F .In particular, GF 6= AFBF .



Conditional permutability

Remark

c-permutability fails to satisfy the property of persistence inintermediate subgroups.

Example

Let G = S4 and let Y ∼= C2 generated by a transposition.

Let V be the normal subgroup of G of order 4 and X a subgroup of Vof order 2, X 6= Z (VY ). Then

X and Y are c-permutable in G

X and Y are not c-permutable in 〈X ,Y 〉.


Permutability properties Complete c-permutability

Complete c-permutability

Definitions

(Guo, Shum, Skiba, 05) Let G be a group and let A and B be subgroupsof G.

A and B are completely c-permutable in G (cc-permutable), ifABg = BgA for some g ∈ 〈A,B〉.A and B are totally completely c-permutable (tcc-permutable) ifevery subgroup of A is completely c-permutable in G with everysubgroup of B.

Totally permutable =⇒6⇐=

Totally completely c-permutable =⇒6⇐=

Totally c-permutable



Complete c-permutability and supersolubility

G = AB, A,B totally c-permutable, GU = AUBU

Corollary

(Guo, Shum, Skiba, 06)

Let G = AB be a product of the tcc-permutable subgroups A andB. If A,B ∈ U , then G ∈ U .

Let G = AB be the product of the tcc-permutable subgroups A andB and let p be a prime. If A,B are p-supersoluble, then G isp-supersoluble.



Total complete c-permutability and saturated formations

Question

Are saturated formations F (of soluble groups) containing U closedunder taking products of totally completely c-permutable subgroups?

Theorem

(Arroyo-Jordá, AJ, Pérez-Ramos, 11)Let F be a saturated formation such that U ⊆ F ⊆ S.Let the group G = G1 · · ·Gr be the product of pairwise permutablesubgroups G1, . . . ,Gr , for r ≥ 2. Assume that Gi and Gj aretcc-permutable subgroups for all i , j ∈ {1, . . . , r}, i 6= j . Then:

If Gi ∈ F for all i = 1, . . . , r , then G ∈ F .If G ∈ F , then Gi ∈ F for all i = 1, . . . , r .




Corollary

(Arroyo-Jordá, AJ, Pérez-Ramos, 11)Let F be a saturated formation such that U ⊆ F ⊆ S.Let the group G = G1 · · ·Gr be the product of pairwise permutablesubgroups G1, . . . ,Gr , for r ≥ 2. Assume that Gi and Gj aretcc-permutable subgroups for all i , j ∈ {1, . . . , r}, i 6= j . Then:

GFi E G for all i = 1, . . . , r .GF = GF1 · · ·GFr .




Question

Is it possible to extend the above results on products of tcc-permutablesubgroups to either non-saturated formations or saturated formationsof non-soluble groups F such that U ⊆ F?

We need a better knowledge of structural properties of products oftcc-permutable groups.


Complete c-permutability Structural properties

Structural properties

Lemma

(AJ, AJ, PR, 11) If 1 6= G = AB is the product of tcc-permutablesubgroups A and B, then there exists 1 6= N E G such that eitherN ≤ A or N ≤ B.

Corollary

(AJ, AJ, PR,11) Let the group 1 6= G = G1 · · ·Gr be the product ofpairwise permutable subgroups G1, . . .Gr , for r ≥ 2. Assume that Giand Gj are tcc-permutable subgroups for all i , j ∈ {1, . . . , r}, i 6= j .

Then there exists 1 6= N E G such that N ≤ Gi for some i ∈ {1, . . . , r}.



Subnormal subgroups

Proposition

(AJ,AJ,MP,PR,13) Let the group G = AB be the product oftcc-permutable subgroups A and B. Then

A′ EEG and B′ EEG.

Corollary

(AJ,AJ,MP,PR,13) Let the group G = G1 · · ·Gr be the product of pairwisepermutable subgroups G1, . . . ,Gr , for r ≥ 2. Assume that Gi and Gjare tcc-permutable subgroups for all i , j ∈ {1, . . . , r}, i 6= j . Then:

G′i EEG, for all i ∈ {1, . . . , r}.



Subnormal subgroups

Proposition

(Maier, 92) If G = AB is the product of totally permutable subgroups Aand B, then A ∩ B ≤ F (G), that is, A ∩ B is a subnormal nilpotentsubgroup of G

Example

The above property is not true for products of tcc-permutable subgroups.

Let G = S3 = AB with the trivial factorization A = S3 and B a 2-Sylowsubgroup of G. This is a product of tcc-permutable subgroups, but:A ∩ B = B is not a subnormal subgroup of G.

Let G = S3 = AB with the trivial factorization A = B = S3. This is aproduct of tcc-permutable subgroups, but: A ∩ B = S3 6∈ N .



Nilpotent residuals

Theorem

(Beidleman, Heineken, 99) Let G = AB be a product of the totallypermutable subgroups A and B. Then:

[AN ,B] = 1 and [BN ,A] = 1.

Example

Let V = 〈a,b〉 ∼= C5 × C5 and C6 ∼= C = 〈α, β〉 ≤ Aut(V ) given by:aα = a−1, bα = b−1; aβ = b, bβ = a−1b−1

Let G = [V ]C be the corresponding semidirect product. Then G = AB is theproduct of the tcc-permutable subgroups A = 〈α〉 and B = V 〈β〉. Notice thatA ∈ U , but

BN = BU = V does not centralize A.



Nilpotent residuals

Theorem

(AJ,AJ,MP,PR,13) Let the group G = AB be the product oftcc-permutable subgroups A and B. Then

AN E G and BN E G.

Corollary

(AJ,AJ,MP,PR,13) Let the group G = G1 · · ·Gr be the product of pairwisepermutable subgroups G1, . . . ,Gr , for r ≥ 2. Assume that Gi and Gjare tcc-permutable subgroups for all i , j ∈ {1, . . . , r}, i 6= j . Then

GNi E G, for all i ∈ {1, . . . , r}.



U-hypercentre

Theorem

(Hauck, PR, MP, 03), (Gállego, Hauck, PR, 08) Let G = AB be a product ofthe totally permutable subgroups A and B. Then:

[A,B] ≤ ZU (G)

or, equivalently, G/ZU (G) = AZU (G)/ZU (G)× BZU (G)/ZU (G).

Example

Let G = [V ]C = AB the product of the tcc-permutable subgroupsA = 〈α〉 and B = V 〈β〉 (under the action aα = a−1, bα = b−1; aβ = b,bβ = a−1b−1). Notice that:

ZU (G) = 1 and G is not a direct product of A and B.



Main theorem

Theorem

(AJ,AJ,MP,PR, 13)Let the group G = AB be the product of tcc-permutable subgroups Aand B. Then:

[A,B] ≤ F (G).

For the proof we have used the CFSG.



Consequences of the main theorem

Corollary

(AJ,AJ,MP,PR, 13) Let the group G = G1 · · ·Gr be the product ofpairwise permutable subgroups G1, . . . ,Gr , for r ≥ 2, and Gi 6= 1 for alli = 1, . . . , r . Assume that Gi and Gj are tcc-permutable subgroups forall i , j ∈ {1, . . . , r}, i 6= j . Let N be a minimal normal subgroup of G.Then:

1 If N is non-abelian, then there exists a unique i ∈ {1, . . . , r} suchthat N ≤ Gi . Moreover, Gj centralizes N and N ∩Gj = 1, for allj ∈ {1, . . . , r}, j 6= i .

2 If G is a monolithic primitive group, then the unique minimalnormal subgroup N is abelian.



Consequences of the main theorem

Corollary

(AJ,AJ,MP,PR, 13) Let the group G = AB be the tcc-permutable productof the subgroups A and B. Then:

If A is a normal subgroup of G, then B acts u-hypercentrally on Aby conjugation. In particular, BU centralizes A.

[AU ,BU ] = 1.


Complete c-permutability Total complete c-permutability and formations

Total complete c-permutability and formations

Theorem

(AJ, AJ, MP, PR, 13) Let F be a saturated formation such that U ⊆ F .Let the group G = G1 · · ·Gr be the product of pairwise permutablesubgroups G1, . . . ,Gr , for r ≥ 2. Assume that Gi and Gj aretcc-permutable subgroups for all i , j ∈ {1, . . . , r}, i 6= j . Then:

If Gi ∈ F for all i = 1, . . . , r , then G ∈ F .

If G ∈ F , then Gi ∈ F for all i = 1, . . . , r .

Corollary

Under the same hypotheses:GFi E G for all i = 1, . . . , r .

GF = GF1 · · ·GFr .


Complete c-permutability Total complete c-permutability and formations

Necessity of saturation

Example

Define the mapping f : P −→ { classes of groups } by setting

f (p) =

{(1,C2,C3,C4) if p = 5(G ∈ A : exp(G) |p − 1) if p 6= 5

Let F = (G ∈ S | H/K chief factor of G⇒ AutG(H/K ) ∈ f (p) ∀p ∈ σ(H/K )).

F is a formation of soluble groups such that U ⊆ F .

Let again G = [V ]C = AB be the product of the tcc-permutable subgroupsA = 〈α〉 and B = V 〈β〉 (under the action aα = a−1, bα = b−1; aβ = b,bβ = a−1b−1). Then:

• A,B ∈ F , but G 6∈ F , since G/CG(V ) ∼= C3 × C2 6∈ f (5).

Modifying the construction of the formation F by setting f (5) = (1,C2,C4,C6):

• G,A ∈ F , but B 6∈ F , since B/CB(V ) ∼= C3 6∈ f (5).


References

References

M. Arroyo-Jordá, P. Arroyo-Jordá, A. Martínez-Pastor and M. D.Pérez-Ramos, On finite products of groups and supersolubility, J.Algebra, 323 (2010), 2922-2934.

M. Arroyo-Jordá, P. Arroyo-Jordá and M. D. Pérez-Ramos, On conditionalpermutability and saturated formations, Proc. Edinburgh Math. Soc., 54(2011), 309-319.

M. Arroyo-Jordá, P. Arroyo-Jordá, A. Martínez-Pastor and M. D.Pérez-Ramos, A survey on some permutability properties on subgroupsof finite groups, Proc. Meeting on group theory and its applications, onthe occasion of Javier Otal’s 60 th birthday, Biblioteca RMI, (2012), 1-11.

M. Arroyo-Jordá, P. Arroyo-Jordá, A. Martínez-Pastor and M. D. Pérez-Ramos, On conditional permutability and factorized groups, Annali diMatematica Pura ed Applicata (2013), DOI 10.1007/S10231-012-0319-1.


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new progress on factorized groups and subgroup permutability · new progress on factorized groups...

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